Question

From the following data for the reaction between \(\mathrm{A}\) and \(\mathrm{B}\) \(\begin{array}{llll}{[\mathrm{A}]} & {[\mathrm{B}]} & \text { initial rate } & \left.(\mathrm{mol}]^{-1} \mathrm{~s}^{-1}\right) \\ \mathrm{mol} 1^{-1} & \mathrm{~mol} & \mathrm{l}^{-1} 300 \mathrm{~K} & 320 \mathrm{~K} \\ 2.5 \times 10^{-4} & 3.0 \times 10^{-5} & 5.0 \times 10^{-4} & 2.0 \times 10^{-3} \\\ 5.0 \times 10^{-4} & 6.0 \times 10^{-5} & 4.0 \times 10^{-3} & \- \\ 1.0 \times 10^{-3} & 6.0 \times 10^{-5} & 1.6 \times 10^{-2} & -\end{array}\) Calculate the rate of the equation. a. \(\mathrm{r}=\mathrm{k}[\mathrm{B}]^{1}\) b. \(\mathrm{r}=\mathrm{k}[\mathrm{A}]^{2}\) c. \(r=k[A]^{2}[B]^{1}\) d. \(\mathrm{r}=\mathrm{k}[\mathrm{A}][\mathrm{B}]\)

Short answer

The rate law is option c: \(r = k[A]^2[B]^1\).

Step-by-step solution

Write the general rate equation

The general rate equation for a reaction \[A + B \rightarrow ext{Products}\]is\[r = k[A]^m[B]^n\],where \(r\) is the rate of reaction, \(k\) is the rate constant, \([A]\) and \([B]\) are the concentrations of the reactants, and \(m\) and \(n\) are the orders of the reaction with respect to \(A\) and \(B\), respectively.

Determine the order of reaction with respect to A

Using the data:- Compare Experiments 1 and 2, where \([B]\) is constant.\[ \text{Experiment 1: } [A] = 2.5 \times 10^{-4}, \text{ rate } = 5.0 \times 10^{-4} \]\[ \text{Experiment 2: } [A] = 5.0 \times 10^{-4}, \text{ rate } = 4.0 \times 10^{-3} \]The rate increases by a factor of 8 when \([A]\) doubles:\[\frac{4.0 \times 10^{-3}}{5.0 \times 10^{-4}} = 8 = 2^m \]Solving for \(m\), we find \(m = 2\).

Determine the order of reaction with respect to B

Using the data:- Compare Experiments 1 and 3, where \([A]\) is constant.\[ \text{Experiment 1: } [B] = 3.0 \times 10^{-5}, \text{ rate } = 5.0 \times 10^{-4} \]\[ \text{Experiment 3: } [B] = 6.0 \times 10^{-5}, \text{ rate } = 1.6 \times 10^{-2} \]The rate increases by a factor of 32 when \([B]\) doubles:\[\frac{1.6 \times 10^{-2}}{5.0 \times 10^{-4}} = 32 = 2^n \]Solving for \(n\), we find \(n = 1\).

Write the specific rate law for the reaction

Now that we have determined the orders:\(m = 2\) for \(A\) and \(n = 1\) for \(B\), the rate law is:\[r = k[A]^2[B]^1\]

Choose the correct option

Compare the calculated rate law \(r = k[A]^2[B]^1\) with the provided options:a. \(r = k[B]^1\) b. \(r = k[A]^2\) c. \(r = k[A]^2[B]^1\) d. \(r = k[A][B]\)Option c matches the calculated rate law.

Key concepts

Reaction Order

In the world of chemical reactions, knowing how fast a reaction occurs is essential. This is where the concept of reaction order comes into play. Reaction order tells us the power to which the concentration of a reactant is raised in the rate law. For our case in the example above, the reaction order for - Reactant A is 2. - Reactant B is 1. This means that the rate of the reaction depends on the concentration of A squared and B to the first power. The total order of the reaction itself is the sum of the orders with respect to each reactant. Thus, this reaction overall is a third-order reaction (2 for A plus 1 for B equals 3). This helps us understand which reactant's concentration changes affect the reaction rate the most.

Rate Constant

The rate constant, often symbolized as \( k \), is a crucial factor in the study of chemical kinetics. It is a proportional constant that links the reaction rate to the concentrations of the reactants raised to the power of their respective reaction orders. In the case of our equation, once you know \( m \) and \( n \), the rate constant can be calculated if the rate and reactant concentrations are known at a specific temperature. The value of \( k \) is specific to the reaction and its conditions like temperature and pressure. - Higher values of \( k \) denote a faster reaction.- Lower values indicate a slower reaction. Remember, \( k \) is incredibly informative because it doesn't change when the concentration of reactants changes; rather, it's specific to the condition under which the reaction is studied.

Chemical Kinetics

Chemical kinetics is the subfield of chemistry that studies the speed or rate at which a chemical reaction proceeds. It not only helps to determine the rate law but also sheds light on the mechanism and steps of the chemical process. The experiment with A and B highlights kinetics by showing how different concentrations affect reaction rates. The four main factors affecting the rate of a chemical reaction include:

• Concentration of reactants: As shown in our solution, the higher the concentration, the faster the reaction up to a point.
• Temperature: Not detailed above, but higher temperatures typically increase reaction rates.
• Surface area: Greater surface area can lead to faster reactions, common in solid reactants.
• Catalysts: Substances that increase the rate of a reaction without being consumed in the process.

Understanding kinetics allows chemists to effectively control reaction rates for various practical applications, from industrial processes to pharmaceuticals.

Concentration Effect

The influence of reactant concentrations on the rate of reaction is known as the concentration effect. Our step-by-step solution highlights this effect by demonstrating how altering the concentration of A and B influences the reaction rate. The rate law equation \( r = k[A]^2[B]^1 \) shows that the rate of the reaction depends heavily on how the concentration of A and B are adjusted. An increase in the concentration of A doubled the reaction rate eightfold under certain conditions and adjusting the concentration of B had a different yet still significant impact. The concentration effect is vital not just for understanding the speed of a single reaction but also for modifying and designing reactions for practical benefits in industries like pharmaceuticals, agriculture, and environmental science.